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Theorem eldm 5190
Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
Hypothesis
Ref Expression
eldm.1  |-  A  e. 
_V
Assertion
Ref Expression
eldm  |-  ( A  e.  dom  B  <->  E. y  A B y )
Distinct variable groups:    y, A    y, B

Proof of Theorem eldm
StepHypRef Expression
1 eldm.1 . 2  |-  A  e. 
_V
2 eldmg 5188 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
31, 2ax-mp 5 1  |-  ( A  e.  dom  B  <->  E. y  A B y )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   E.wex 1599    e. wcel 1804   _Vcvv 3095   class class class wbr 4437   dom cdm 4989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-dm 4999
This theorem is referenced by:  dmi  5207  dmcoss  5252  dmcosseq  5254  dminss  5410  dmsnn0  5463  dffun7  5604  dffun8  5605  fnres  5687  opabiota  5921  fndmdif  5976  dff3  6029  frxp  6895  suppvalbr  6907  reldmtpos  6965  dmtpos  6969  aceq3lem  8504  axdc2lem  8831  axdclem2  8903  fpwwe2lem12  9022  nqerf  9311  shftdm  12885  xpsfrnel2  14943  bcthlem4  21743  dchrisumlem3  23652  eupath  24957  fundmpss  29171  elfix  29528  fnsingle  29544  fnimage  29554  funpartlem  29567  dfrdg4  29575  prtlem16  30585
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